Geophysical Research Letters
Supporting Information for
A Strong Seasonal Dependence in the Martian Hydrogen Exosphere
Dolon Bhattacharyya1, John T. Clarke1, Jean-Loup Bertaux2, Jean-Yves Chaufray2, Majd
Mayyasi1
1Center
for Space Physics, Boston University, Boston MA, USA
2LATMOS/CNRS
Guyancourt, France
Contents of this file
Text S1 to S2
Figures S1 to S2
Introduction



A description of the reduction process for the HST data and determination of its
absolute calibration;
Description of the radiative transfer model with relevant equations;
Discussion of the goodness of the model fits to the data;
S1. Reduction of the HST Observations
HST images of Mars in the far ultraviolet (FUV) were obtained using the Advanced
Camera for Surveys (ACS) Solar Blind Channel (SBC) instrument in 2007 and 2014. These
images have been reduced by custom procedures developed at Boston University (BU)
rather than using the Space Telescope Science Institute’s (STScI) routines. A detailed
description of the BU pipeline can be found in Clarke et al. (2009). This pipeline has been
adapted for the Mars observations as described in Clarke et al. (2014).
S1.1. Absolute Calibration for ACS-SBC at Lyman α
1
It is essential to determine the total throughput which includes the optical telescope
assembly (OTA) and the ACS-SBC instrument at Lyman α to calculate the absolute
calibration. The throughput at UV wavelengths for HST has been modeled by studying
the instrument response and characteristics at those wavelengths and is reproduced in
Figure 1 (Avila et al., 2015). This curve is based on bandpass-averaged count rates
looking at flux standard calibration stars, and the shape of the curve is not directly
measured but based on assumed component efficiencies. From this figure it was
determined that the total throughput of the system (OTA+ACS) at Lyman α has a value
of 0.0498. By knowing this value and multiplying it with the total collecting area of HST
and the solid angle of 1 pixel of the detector used to image Mars, the total number of
Lyman α photon counts which correspond to 1 kilo-Rayleigh (kR) has been calculated
and has a value of 0.002633 counts/pixel-sec-kR.
S2. Modeling the HST observations
The resonantly scattered Lyman α photons from the exosphere of Mars have been
modeled using a radiative transfer model based on Chaufray et al. (2008), and the model
has been tested by comparison with the Chaufray et al. (2008) model output under
similar conditions of exospheric temperature and density values for hydrogen at Mars.
There are two free parameters in the model, temperature and number density of H at the
exobase. The density profile of hydrogen is derived using a simple diffusion model
described in Chaufray et al. (2008) below the exobase. Above the exobase it follows the
characteristics of a Chamberlain exosphere.
S2.1 Radiative Transfer Model
The model is based on the assumptions of Thomas (1963) which includes an isothermal
atmosphere with uniform density resulting in spherical symmetry, a Maxwell-Boltzmann
velocity distribution for the H atoms with Complete Frequency Redistribution. The model
also assumes a Gaussian-Dopplerian absorption profile for the H atoms as well as a flat
solar line resulting in the H atoms only absorbing photons at the central wavelength of
the Lyman α line (121.57 nm). Under these assumptions, the integrated intensity I, along
a particular line of sight is given by
𝐼(𝑟, Ω) =
𝑃(𝜃)
4𝜋
∫(𝜀0 (𝑟) 𝑇(𝜏) 𝑒 −𝜏𝐶𝑂2 +
1
4𝜋
𝜀𝑛 (𝑟) 𝑇(𝜏) 𝑒 −𝜏𝐶𝑂2 ) 𝑑𝑟
(S1)
where 𝑃(𝜃) is the scattering phase function with 𝜃 as the scattering angle and has the
form
𝑃(𝜃) =
11
12
+
1
4
𝑐𝑜𝑠 2 𝜃
(S2)
2
𝜏 and 𝜏𝐶𝑂2 represent the optical thickness in Lyman α for H and CO2 respectively.
𝜀0 (𝑟) and 𝜀𝑛 (𝑟)are the volume emission rates due to single and multiple scattering in
the atmosphere of Mars in photons/cm3/sec and have the following forms
𝜀0 (𝑟) = 𝑔 × 𝑛𝐻 (𝑟) × 𝑇(𝜏𝑠𝑜𝑙 ) × 𝑒 −𝜏𝐶𝑂2
𝑑Ω
∞
𝜀𝑛 (𝑟) = 𝑔 × 𝜎0 × 𝑛𝐻 (𝑟) × ∫ 4𝜋 ∫𝑟 𝑆(𝑟 ′ ) 𝐺(𝜏) 𝑒 −𝜏𝐶𝑂2 𝑑𝑟′
(S3)
(S4)
Here 𝑛𝐻 (𝑟) is the number density and 𝜎0 the Lyman α scattering cross section of H,
𝑇(𝜏)and 𝐺(𝜏) are the Holstein functions (Holstein, 1947), 𝑆(𝑟 ′ ) is the multiple scattering
𝜀(𝑟)
source function (
𝑔
) , 𝜏 is the line of sight optical depth and 𝑔 is the H solar Lyman α
scattering frequency in sec-1. The 𝑔 value at Mars was calculated from the solar Lyman α
flux at Earth, adjusted for solar rotation and distance to Mars (Rottmann et al., 2006;
Emerich et al., 2005). The solution to equation (S4) is based upon an iterative approach
described in Quemerais and Bertaux (1993).
S2.2. Analysis of the HST Observations and description of the 𝝌𝟐 values for the fits
Different combinations of temperature and density ranging from 170 - 440 K and 1 x 104
– 5 x 105 cm-3 have been used to model the exosphere of Mars. The first modeling
approach assumed a single maxwellian distribution of hydrogen present in the
exosphere of Mars. The best fit temperature density for each observation was
determined by minimizing the reduced 𝝌𝟐 for the model fits to the data.
𝜒 2 (𝑇𝑒𝑥𝑜 , 𝑛𝑒𝑥𝑜 ) =
1
𝑛−2
∑𝑛𝑖=1
(𝐼𝑑𝑎𝑡𝑎,𝑖 (𝑟)− 𝐼𝑚𝑜𝑑𝑒𝑙,𝑖 (𝑟))2
2
𝜎𝑑𝑎𝑡𝑎,𝑖
(𝑟)
(S5)
Here Idata and Imodel represent the intensity of the data and model at a particular radial
distance and the σ in the denominator represents the uncertainty in the data at a
particular radial distance. The n in the denominator represents the total number of data
points while the 2 is a result of the two free parameters of the model. The total number
of data points for a particular observation is dependent upon the plate scale of the
image, a 2000 x 2000 pixel picture. The total number of pixels that fall on a line drawn
from the center of Mars in the image passing through the sub-solar point up to the edge
of the image on the dayside, determines n. This value of n usually ranges from 1100 –
1400 points which when multiplied by the plate scale for the image gives the radial
distances at which intensity measurements have been taken. In determining the
uncertainty in the denominator of equation (S5), we have not taken into account the
uncertainty due to the modeling process, which drives the reduced 𝝌𝟐 values above 1,
owing to the difficulty in accurately quantifying the inaccuracies in the modeling process
due to certain inherent assumptions. These include a spherically symmetric density
distribution as there could very well be regions enhanced density as found on the
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nightside of Venus (Grebowsky et al., 1996; Hartle et al., 1996). Other assumptions like a
single maxwellian distribution of hydrogen atoms might not be applicable for the
exosphere of Mars as it is well known that a superthermal population exists at Venus
which also has a dominant CO2 atmosphere (Kumar et al., 1978; Anderson, 1976; Bertaux,
et al., 1977; Chaufray et al., 2012). Research into these characteristics of the hydrogen
exosphere at Mars is currently ongoing with the MAVEN spacecraft.
The best temperature values to the HST observations assume a single maxwellian
distribution for the hydrogen population. These values range from 360 – 440 K. Except
for 27th October 2007, 9th November 2007 and 30th May 2014, all the other observations
have 440 K as the best fit temperature which is at the maximum boundary of the
temperature range used to fit the data. This probably indicates that a true minimum in
temperature and density has not been achieved for these observations and the best fit
temperature for these days is most likely higher than 440 K. However, temperatures of
the exosphere higher than 440 K are not supported by our understanding of the energy
balance of the upper atmosphere from martian global circulation models (Chaufray et al.,
2015; Bougher et al., 1999, 2009). Therefore we consider the possibility of the presence of
a two-component population of hydrogen with one component at a lower temperature
while the other at a much higher temperature as has been observed for Venus (Kumar et
al., 1978; Anderson, 1976; Bertaux et al., 1977; Chaufray et al., 2012). Such a possibility
would increase the temperature of the entire hydrogen exosphere as well as broaden the
martian hydrogen Lyman α line. The best fit model profiles to the data using both the
single and the two-component population of hydrogen are presented in Figure S2a, b, c
and d. From the reduced 𝝌𝟐 values presented in Table 2 in the paper, for the model fits
using both approaches, it appears as though the single component model fits the data
better on occasions when Mars is further away from the Sun, whereas the twocomponent model gives a better fit to the data for occasions when Mars is closer to the
Sun in its orbit. This may imply that the production of the hot component is tied to the
solar EUV flux, but will require further analysis.
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Figure S1. Total throughput of HST’s optical telescope assembly and the ACS instrument
for ultraviolet wavelengths.
6
Figure S2a: Best model fits to the HST 2014 data using the one component model
assumption
Figure S2b: Best model fits to the HST 2014 data using the two component model
assumption.
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Figure S2c: Best model fits to the HST 2007 data for the one component model
assumption
Figure S2d: Best model fits to the HST 2007 data for the two component model
assumption.
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